#423987

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(2,4)(9,10,13,12)(11,15,16,14)(17,18,20,23)(19,26,27,22)(21,24,29,25)(28,31,32,30);(5,6)(7,8)(9,11,13,16)(10,14,12,15)(17,19,20,27)(18,22,23,26)(21,28,29,32)(24,30,25,31);(1,2,3,4)(6,8)(18,23)(22,26)(24,25)(30,31);(9,12,13,10)(11,14,16,15)(18,24)(21,29)(22,30)(23,25)(26,31)(28,32)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fi,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331183);